Is there hard evidence that the grant peer review system performs significantly better than random?

No such study exists.

You have to realize that the current model of funding research through grants is rather recent. Fifty years ago, research was funded with recurring credits: a lab/researcher got a fixed amount of money each year to do research as they saw fit. Only applied researchers got industrial grants to develop precise new applications, but this wasn't the core of their funding.

The current system arose due to the combination of two factors:

• The growing hegemony of the neo-liberal ideology in the 80s-90s and the commodification of society as a whole. According to this ideology, research as a whole is supposedly more efficient (for an ill-defined notion of "efficient") if researchers compete with one another on the global market of research funding. They will "naturally" tend to research what works in order to keep their funding.
• Politicians' will to govern research. Research, and especially fundamental research, is a notoriously difficult thing to govern: you never know where an idea might lead, or what new ideas will come up along the way. You don't know what will work and what won't. You don't know what will be applicable and what will remain a nice theoretical idea. Otherwise it wouldn't be research.
  But rulers want their word in this: they want to tell their voters that their policies improved lives in a very direct way as a result of such and such decision they took. This is why words like "governance" and "steering" are becoming more and more prevalent in modern academia: administrators are not in the business of just helping researchers carry out their research, they are in the business of ensuring that researchers are researching what they are supposed to. And how do you force someone to do what you want, short of physical violence? Through economical violence.

Of course, this yields absurd situations, such as the fabled year when the French National Research Agency spent more on administration than on research. And such thinking paves the way for short-term research with zero long term goals, and especially not goals reachable in more than five years (the usual length of "long-term" grants). But these decisions are not driven by rational thought; they are driven by ideology.

I don't know about such studies, but I have served on ~20 panels to review proposals. While I'm entirely willing to believe that multiple panels will not agree on the relative ordering of proposals, I am quite convinced that they will in general agree on which proposals are "good" and which are "not good".

In any given round, a panel (at the National Science Foundation) will review on the order of 20 proposals. What may not be obvious to outsiders is that of these 10 are pretty obviously not fundable. 3-5 are eventually rated as excellent and the remainder as "pretty good". I am pretty convinced that if you ran multiple panels on these 20 proposals, that this classification into three groups will be more or less stable. What I am also convinced of is that panels will produce different rankings within each group.

The consequence is that it is quite possible that different panels will results in a different set of proposals being funded, given that only ~4 proposals out of the 20 will be funded. But overall, the result will still be far from random. The top 3-5 are still likely to be funded, the bottom 10 will definitely not be funded, and it's a toss-up in between.

There is another aspect to the question at hand. A random assignment of monies to projects can be expected to fail because the system in place, whatever it is, induces certain behaviors. One wants to design a system so that positive behavior is encouraged and negative behavior discouraged.

In a system of peer review those seeking funds are induced to give their analysis of why their proposed project should be funded and why it is highly likely to be successful. They know that their words will have consequences and that the review will be carried out by knowledgeable people who have assumed a certain responsibility.

But if the system were random, and known to be random, the behavior induced would be quite different. There would be no real need to spend the time and effort to examine and explain the background and methodology to be used and to guarantee that it meets scientific rigor. The consequence would be that many more proposals would be made and most of them would be junk.

With that background, note that the proposals that are actually made to funding agencies are pretty much all guaranteed to be at least adequate. So the peer review boards are choosing from among the best proposals that might be expected, rather from the trash that might be thrown out by charlatans.

This brings me to a thought experiment that I've used in other contexts. Suppose, in a system that repeats, you rank a set of things on some linear scale from best to worst, and you discard the worst. The worst is replaced. This is sometimes what happens in companies that rank employees on some numeric scale and fire the worst performer.

But, if you started out with the, say, ten best performers in the world, your system will only worsen your overall system, since everyone else will be worse than the ten you start with.

Thus the question asked by the OP doesn't occur in a vacuum. Nor, I suggest, can the various proposals be ranked in theory on a best to worst scale. There are too many variables and something better on one scale isn't as good on another. So, while the system has flaws, it is in the nature of the world that some flaws will exist. But a system that induces and rewards good behavior is, in principle, about as good as we can hope for.

The various judgments made by the review boards are, in the larger sense of things, relatively minor.